Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Provide solution for RD Sharma maths Class 12 Chapter 19 Definite Integrals Exercise 19.4 (a) Question 8 textbook solution.

Provide solution for  RD Sharma maths Class 12 Chapter 19 Definite Integrals Exercise 19.4 (a) Question 8 textbook solution.

Answers (1)

Answer : \frac{\pi}{4}

Given : \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{\cos ^{2} x}{1+e^{x}} d x

Hint : the formula of  a of \int_{a}^{b} f(x) d x and then add the two equations

Solution : I=\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{\cos ^{2} x}{1+e^{x}} d x \quad-----(1)

\begin{aligned} &I=\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{\cos ^{2}(-x)}{1+e^{-x}} d x \\ &I=\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{\cos ^{2} x}{1+e^{-x}} d x \end{aligned}

\begin{aligned} &\int_{a}^{b} f(x) d x=\int_{a}^{b} f(a+b-x) d x \\ &I=\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{e^{x} \cos ^{2} x}{e^{x}+1} d x \quad-----(2) \end{aligned}

Add (1) and (2)

\begin{aligned} &2 I=\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{\cos ^{2} x+e^{x} \cos ^{2} x}{1+e^{x}} d x \\ &2 I=\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \cos ^{2} x d x \end{aligned}

\begin{aligned} &2 I=\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{1+\cos 2 x}{2} d x\left[\cos 2 x=2 \cos ^{2} x-1\right] \\ &I=\frac{1}{4}\left(x+\frac{\sin 2 x}{2}\right)_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \\ &I=\frac{1}{4}\left[\left(\frac{\pi}{2}+0\right)-\left(\frac{-\pi}{2}+0\right)\right] \end{aligned}

\begin{aligned} &I=\frac{1}{4}\left(\frac{\pi}{2}+\frac{\pi}{2}\right) \\ &I=\frac{\pi}{4} \end{aligned}

Posted by

infoexpert23

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

CBSE announces deadline for Class 10, 12 direct admissions, subject change
anam-khan

CBSE warns about unauthorised sources offering ‘quick’ ways of getting duplicate marksheets, certificates
anam-khan

CBSE unveils 8 major changes for Class 10, 12 board exams; what students need to know

Latest Question